International Journal for Numerical Methods in Engineering

Published by Wiley
Online ISSN: 1097-0207
Print ISSN: 0029-5981
Assessing the risk of rupture of intracranial aneurysms is important for clinicians because the natural rupture risk can be exceeded by the small but significant risk carried by current treatments. To this end numerous investigators have used image-based computational fluid dynamics models to extract patient-specific hemodynamics information, but there is no consensus on which variables or hemodynamic characteristics are the most important. This paper describes a computational framework to study and characterize the hemodynamic environment of cerebral aneurysms in order to relate it to clinical events such as growth or rupture. In particular, a number of hemodynamic quantities are proposed to describe the most salient features of these hemodynamic environments. Application to a patient population indicates that ruptured aneurysms tend to have concentrated inflows, concentrated wall shear stress distributions, high maximal wall shear stress and smaller viscous dissipation ratios than unruptured aneurysms. Furthermore, these statistical associations are largely unaffected by the choice of physiologic flow conditions. This confirms the notion that hemodynamic information derived from image-based computational models can be used to assess aneurysm rupture risk, to test hypotheses about the mechanisms responsible for aneurysm formation, progression and rupture, and to answer specific clinical questions.
This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double-sided node-to-surface contact or self-contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time-stepping scheme. The time step may be fixed or time-adaptive. New demands on global collision detection are discussed exemplified by position codes and node-to-segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.
A two-step explicit FEM solution algorithm for the three-dimensional compressible Euler and Navier-Stokes equations based on unstructured triangular and tetrahedral grids is described and demonstrated. The method represents an extension and refinement of the algorithms presented by Loehner et al. (1984 and 1985), Peraire et al. (1987), and Morgan et al. (1987). The formulation and numerical implementation are outlined; the mesh generation, data structures, and adaptive remeshing are explained; and results for a two-dimensional airfoil, a three-dimensional engine air intake, a B747 in landing configuration, and a generic fighter aircraft are presented in extensive graphics and discussed in detail.
Abel's integral equation is the governing equation for certain problems in physics and engineering, such as radiation from distributed sources. The finite element method for the solution of this non-linear equation is presented for problems with cylindrical symmetry and the extension to more general integral equations is indicated. The technique was applied to an axisymmetric glow discharge problem and the results show excellent agreement with previously obtained solutions
The numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries. This point is critical for the analysis of wave propagation in heterogeneous or layered solids. Various techniques such as Absorbing Boundary Conditions, infinite elements or Absorbing Boundary Layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this paper, a simple absorbing layer method is proposed: it is based on a Rayleigh/Caughey damping formulation which is often already available in existing Finite Element softwares. The principle of the Caughey Absorbing Layer Method is first presented (including a rheological interpretation). The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences. A comparison with the PML method is first performed for pure P-waves and the method is shown to be reliable in a more complex 2D case involving various wave types and incidences. It may thus be used for various types of problems involving elastic waves (e.g. machine vibrations, seismic waves, etc).
An advanced automated design procedure for minimum weight design of structures (ACCESS 2) is reported. Design variable linking, constraint deletion, and explicit constraint approximation are used to effectively combine finite element and nonlinear mathematical programming techniques. The approximation concepts approach to structural synthesis is extended to problems involving fiber composite structure, thermal effects and natural frequency constraints in addition to the usual static stress and displacement limitations. Sample results illustrating these new features are given.
Accuracies of solutions (structural temperatures and thermal stresses) obtained from different thermal and structural FEMs set up for the Space Shuttle Orbiter (SSO) are compared and discussed. For studying the effect of element size on the solution accuracies of heat-transfer and thermal-stress analyses of the SSO, five SPAR thermal models and five NASTRAN structural models were set up for wing midspan bay 3. The structural temperature distribution over the wing skin (lower and upper) surface of one bay was dome shaped and induced more severe thermal stresses in the chordwise direction than in the spanwise direction. The induced thermal stresses were extremely sensitive to slight variation in structural temperature distributions. Both internal convention and internal radiation were found to have equal effects on the SSO.
The accuracy of a new class of concurrent procedures for transient finite element analysis is examined. A phase error analysis is carried out which shows that wave retardation leading to unacceptable loss of accuracy may occur if a Courant condition based on the dimensions of the subdomains is violated. Numerical tests suggest that this Courant condition is conservative for typical structural applications and may lead to a marked increase in accuracy as the number of subdomains is increased. Theoretical speed-up ratios are derived which suggest that the algorithms under consideration can be expected to exhibit a performance superior to that of globally implicit methods when implemented on parallel machines.
A second-order method for solving two-point boundary value problems on a uniform mesh is presented where the local truncation error is obtained for use with the deferred correction process. In this simple finite difference method the tridiagonal nature of the classical method is preserved but the magnitude of each term in the truncation error is reduced by a factor of two. The method is applied to a number of linear and nonlinear problems and it is shown to produce more accurate results than either the classical method or the technique proposed by Keller (1969).
A modified Green operator is proposed as an improvement of Fourier-based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite. Furthermore, it is shown that the method produces much more accurate local fields inside highly-conducting and quasi-insulating phases, as well as in the vicinity of the phases interfaces. These good properties stem from the discretization of Green's function, which is consistent with the pixel grid while retaining the local nature of the operator that acts on the polarization field. Finally, a fast implementation of the "direct scheme" of Moulinec et al. (1994) that allows for parcimonious memory use is proposed.
Several explicit integration algorithms with self-adative time integration strategies are developed and investigated for efficiency and accuracy. These algorithms involve the Runge-Kutta second order, the lower Runge-Kutta method of orders one and two, and the exponential integration method. The algorithms are applied to viscoplastic models put forth by Freed and Verrilli and Bodner and Partom for thermal/mechanical loadings (including tensile, relaxation, and cyclic loadings). The large amount of computations performed showed that, for comparable accuracy, the efficiency of an integration algorithm depends significantly on the type of application (loading). However, in general, for the aforementioned loadings and viscoplastic models, the exponential integration algorithm with the proposed self-adaptive time integration strategy worked more (or comparably) efficiently and accurately than the other integration algorithms. Using this strategy for integrating viscoplastic models may lead to considerable savings in computer time (better efficiency) without adversely affecting the accuracy of the results. This conclusion should encourage the utilization of viscoplastic models in the stress analysis and design of structural components.
This work presents a method to adaptively refine reduced-order models a posteriori without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive $h$-refinement: it enriches the reduced-basis space by `splitting' selected basis vectors into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed via recursive $k$-means clustering of the state variables using snapshot data. The method identifies the vectors to split using a dual-weighted residual approach that seeks to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or high-fidelity solves. Further, it enables the reduced-order model to satisfy any prescribed error tolerance online regardless of its original fidelity, as a completely refined reduced-order model is equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.
The application of adaptive finite element methods to the solution of transient heat conduction problems in two dimensions is investigated. The computational domain is represented by an unstructured assembly of linear triangular elements and the mesh adaptation is achieved by local regeneration of the grid, using an error estimation procedure coupled to an automatic triangular mesh generator. Two alternative solution procedures are considered. In the first procedure, the solution is advanced by explicit timestepping, with domain decomposition being used to improve the computational efficiency of the method. In the second procedure, an algorithm for constructing continuous lines which pass only once through each node of the mesh is employed. The lines are used as the basis of a fully implicit method, in which the equation system is solved by line relaxation using a block tridiagonal equation solver. The numerical performance of the two procedures is compared for the analysis of a problem involving a moving heat source applied to a convectively cooled cylindrical leading edge.
In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.
The present paper describes recent advances and trends in finite element developments and applications for solidification problems. In particular, in comparison to traditional methods of approach, new enthalpy-based architectures based on a generalized trapezoidal family of representations are presented which provide different perspectives, physical interpretation and solution architectures for effective numerical simulation of phase change processes encountered in solidification problems. Various numerical test models are presented and the results support the proposition for employing such formulations for general phase change applications.
The 'transfinite element method' (TFEM) proposed by Tamma and Railkar (1987 and 1988) for the analysis of linear and nonlinear heat-transfer problems is described and demonstrated. The TFEM combines classical Galerkin and transform approaches with state-of-the-art FEMs to obtain a flexible hybrid modeling scheme. The fundamental principles of the TFEM and the derivation of the governing equations are reviewed, and numerical results for sample problems are presented in extensive graphs and briefly characterized. Problems analyzed include a square plate with a hole, a rectangular plate with natural and essential boundary conditions and varying thermal conductivity, the Space Shuttle thermal protection system, a bimaterial plate subjected to step temperature variations, and solidification in a semiinfinite liquid slab.
The introduction of composite materials is having a profound effect on the design process. Because these materials permit the designer to tailor material properties to improve structural, aerodynamic and acoustic performance, they require a more integrated multidisciplinary design process. Because of the complexity of the design process numerical optimization methods are required. The present paper is focused on a major difficulty associated with the multidisciplinary design optimization process—its enormous computational cost. We consider two approaches for reducing this computational burden: (i)development of efficient methods for cross-sensitivity calculation using perturbation methods; and (ii) the use of approximate numerical optimization procedures. Our efforts are concentrated upon combined aerodynamic-structural optimization. Results are presented for the integrated design of a sailplane wing. The impact of our computational procedures on the computational costs of integrated designs is discussed.
Intensive research and development in Computational Fluid Dynamics (CFD) has recently produced many powerful CFD codes to simulate complex aerodynamic phenomena. However, in order to enhance the usefulness of these CFD codes for design practice, development of design sensitivity equations compatible to these codes becomes very important. This paper represents a part of such an effort to develop a sensitivity analysis methodology that enables the sensitivity equations to be implemented into existing CFD codes with minimal code modification. The methodology is based upon a preelimination procedure which accounts for consistently linearized boundary conditions. Formulations of both the direct differentiation and the adjoint variable methods will be presented in the paper.
A complete mathematical model is formulated to analyze the effects of mean-flow incidence angle on the unsteady aerodynamics of an oscillating airfoil in an incompressible flow field. A velocity potential formulation is utilized. The steady flow is independent of the unsteady flow field but coupled to it through the boundary conditions on the oscillating airfoil. The numerical solution technique for both the steady and unsteady flow fields is based on a locally analytical method. The flow model and solution method are then verified through the excellent correlation obtained with the Theodorsen oscillating-flat-plate and Sears transverse-gust classical solutions. The effects of mean flow incidence on the steady and oscillating airfoil aerodynamics are then investigated.
A generalized curvilinear coordinate Taylor weak statement implicit finite element algorithm is developed for the two-dimensional and axisymmetric compressible Navier-Stokes equations for ideal and reacting gases. For accurate hypersonic simulation, air is modeled as a mixture of five perfect gases, i.e., molecular and atomic oxygen and nitrogen as well as nitric oxide. The associated pressure is then determined via Newton solution of the classical chemical equilibrium equation system. The directional semidiscretization is achieved using an optimal metric data Galerkin finite element weak statement, on a developed 'companion conservation law system', permitting classical test and trial space definitions. Utilizing an implicit Runge-Kutta scheme, the terminal algorithm is then nonlinearly stable, and second-order accurate in space and time on arbitrary curvilinear coordinates. Subsequently, a matrix tensor product factorization procedure permits an efficient numerical linear algebra handling for large Courant numbers. For ideal- and real-gas hypersonic flows, the algorithm generates essentially nonoscillatory numerical solutions in the presence of strong detached shocks and boundary layer-inviscid flow interactions.
In this article, various numerical methods to solve multicontact problems within the nonsmooth discrete element method are presented. The techniques considered to solve the frictional unilateral conditions are based both on the bipotential theory introduced by G. de Saxcé and the augmented Lagrangian theory introduced by P. Alart. Following the ideas of Z.-Q. Feng a new Newton method is developed to improve these classical algorithms, and numerical experiments are presented to show that these methods are faster than the previous ones, provide results with a better quality, and are less sensitive to the numerical parameters. Moreover, a stopping criterion that ensures a good mechanical property of the solution is provided.
The mathematical structure underlying the rate equations of a recently-developed constitutive model for the coupled viscoplastic-damage response of anisotropic composites is critically examined. In this regard, a number of tensor projection operators have been identified, and their properties were exploited to enable the development of a general computational framework for their numerical implementation using the Euler fully-implicit integration method. In particular, this facilitated (i) the derivation of explicit expressions of the (consistent) material tangent stiffnesses that are valid for both three-dimensional as well as subspace (e.g. plane stress) formulations, (ii) the implications of the symmetry or unsymmetry properties of these tangent operators from a thermodynamic standpoint, and (iii) the development of an effective time-step control strategy to ensure accuracy and convergence of the solution. In addition, the special limiting case of inviscid elastoplasticity is treated. The results of several numerical simulations are given to demonstrate the effectiveness of the schemes developed.
A generalized formulation of the Energy-Momentum Methodwill be developed within the framework of the Generalized-α Methodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response. Furthermore, the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the Constraint Energy-Momentum Algorithm. The goal of this general approach of implicit energy-conserving and decaying time integration schemes is, to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties. Numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non-linear structural dynamics. Among the methods considered will be the Newmark Method, the classical α-methods, the Energy-Momentum Methodwith and without numerical dissipation, the Constraint Energy-Momentum Algorithm and the Constraint Energy Method. Copyright © 1999 John Wiley & Sons, Ltd.
The performance of group implicit algorithms is assessed on actual concurrent computers. It is shown that, as the number of subdomains is increased, performance enhancements are derived from two sources: the increased parallelism in the computations; and a reduction in equation solving effort. Moreover, these two performance enhancements are synergistic, in the sense that the corresponding speed-ups are multiplied, rather than merely added. Simulations on a 32-node hypercube are presented for which the interprocessor communications efficiencies obtained are consistently in excess of 90 percent.
A family of hierarchical algorithms for nonlinear structural equations are presented. The algorithms are based on the Davidenko-Branin type homotopy and shown to yield consistent hierarchical perturbation equations. The algorithms appear to be particularly suitable to problems involving bifurcation and limit point calculations. An important by-product of the algorithms is that it provides a systematic and economical means for computing the stepsize at each iteration stage when a Newton-like method is employed to solve the systems of equations. Some sample problems are provided to illustrate the characteristics of the algorithms.
Non-linear programming algorithms play an important role in structural design optimization. Fortunately, several algorithms with computer codes are available. At NASA Lewis Research Centre, a project was initiated to assess the performance of eight different optimizers through the development of a computer code CometBoards. This paper summarizes the conclusions of that research. CometBoards was employed to solve sets of small, medium and large structural problems, using the eight different optimizers on a Cray-YMP8E/8128 computer. The reliability and efficiency of the optimizers were determined from the performance of these problems. For small problems, the performance of most of the optimizers could be considered adequate. For large problems, however, three optimizers (two sequential quadratic programming routines, DNCONG of IMSL and SQP of IDESIGN, along with Sequential Unconstrained Minimizations Technique SUMT) outperformed others. At optimum, most optimizers captured an identical number of active displacement and frequency constraints but the number of active stress constraints differed among the optimizers. This discrepancy can be attributed to singularity conditions in the optimization and the alleviation of this discrepancy can improve the efficiency of optimizers.
An element stiffness matrix can be derived by the conventional potential energy principle and, indirectly, also by generalized variational principles, such as the Hu-Washizu principle and the Hellinger-Reissner principle. The present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the Hellinger-Reissner principle. It is found that the resulting scheme is equivalent to that considered by Tong (1982) for the construction of hybrid stress elements. In Tong's scheme the inversion of a large flexibility matrix can be avoided. It is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the Hellinger-Reissner principle and the Hu-Washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method.
A review is given of recent advances in two aspects of the numerical simulation of the buckling and postbuckling responses of composite structures. The first aspect is exploiting non-traditional symmetries exhibited by composite structures; and strategies for reducing the size of the model and the cost of the buckling and postbuckling analyses in the presence of symmetry-breaking conditions (e.g., asymmetry of the material, geometry, and/or loading). The second aspect pertains to the prediction of onset of local delamination in the postbuckling range and accurate determination of transverse shear stresses in the structure. The accuracy and effectiveness of the strategies developed are demonstrated by means of a numerical example.
Boundary conditions of the Schweitzer–Cirak type.
Cylindrical shell subject to point loads. Left: Undeformed reference configuration. Middle: Control mesh (16 × 24) of the quasistatic solution at maximum load P = 4 × 104, without ghost elements. Right: Limit surface of the same configuration, with the stretching energy density on a logarithmic color scale.
Comparison between a crumpled shell (top row) and a growing shell in spherical confinement (bottom row). Network vertices are numbered for easy identification.
A thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff-Love theory of thin shells is derived and extended to allow for arbitrary in-plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasi-static, elastic limit.
An efficient preconditioned conjugate gradient (PCG) technique and a computational procedure are presented for the analysis of symmetric anisotropic structures. The technique is based on selecting the preconditioning matrix as the orthotropic part of the global stiffness matrix of the structure, with all the nonorthotropic terms set equal to zero. This particular choice of the preconditioning matrix results in reducing the size of the analysis model of the anisotropic structure to that of the corresponding orthotropic structure. The similarities between the proposed PCG technique and a reduction technique previously presented by the authors are identified and exploited to generate from the PCG technique direct measures for the sensitivity of the different response quantities to the nonorthotropic (anisotropic) material coefficients of the structure. The effectiveness of the PCG technique is demonstrated by means of a numerical example of an anisotropic cylindrical panel.
This paper describes new and recent advances in the development of a hybrid transfinite element computational methodology for applicability to conduction/convection/radiation heat transfer problems. The transfinite element methodology, while retaining the modeling versatility of contemporary finite element formulations, is based on application of transform techniques in conjunction with classical Galerkin schemes and is a hybrid approach. The purpose of this paper is to provide a viable hybrid computational methodology for applicability to general transient thermal analysis. Highlights and features of the methodology are described and developed via generalized formulations and applications to several test problems. The proposed transfinite element methodology successfully provides a viable computational approach and numerical test problems validate the proposed developments for conduction/convection/radiation thermal analysis.
This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation-diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non-linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non-negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always non-negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice-versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response.
A four-node element divided into four smoothing cells. 
A sample quadrilateral.
A parallelogram shaped element
Geometry and loads for a cantilever beam.
The convergence of the numerical energy to the exact energy with irregular meshes (α ir = 0.2) for the cantilever beam problem.
This letter aims at resolving the issues raised in the recent short communication [1] and answered by [2] by proposing a systematic approximation scheme based on non-mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) [3, 4, 5, 6, 7, 8, 9] and resolve the existence, linearity and positivity deficiencies pointed out in [1]. We show that Wachspress interpolants [10] computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results which are almost identical to those of the SFEM initially proposed in [3]. These results that the proposed approximation scheme forms a strong and rigorous basis for construction of smoothed finite element methods.
This paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen-Lo$\grave{e}$ve expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by $k-$version/$p-$version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We show that the derived a priori error estimates are uniform with respect to the Lam$\acute{e}$ constant $\lambda\in (0, +\infty)$. Finally, we provide some numerical results.
An optimum design sensitivity analysis capability is reported which exploits the approximation concepts-dual method formulation of the minimum weight structural sizing problem. An efficient iterative solution technique is used to facilitate determination of sensitivity derivatives for both primal and dual variables. Estimates on the useful range of parameter perturbations, over which the optimum design sensitivity projections can be expected to yield satisfactory revised optimum designs, are also obtained. Numerical results for several example problems will be presented to illustrate the effectiveness of the capability reported.
The application of second derivative information for solving structural optimization problems is considered. In the present method, rather than building approximate nonlinear forms for the objective function and constraints, only linear approximations are used. A separable quadratic approximation of the Lagrangian function is included in the subproblem statement. The method has been successfully used for simple problems that can be solved in closed form, in addition to the sizing optimization of trusses, and it is shown to converge faster than the convex linearization method or the method of moving asymptotes.
A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.
Differentiating matrices allow the numerical differentiation of functions defined at points of a discrete grid. Previous derivations of these matrices have been restricted to grids with uniformly spaced points, and the resulting derivative approximations have lacked precision, especially at endpoints. The present work derives differentiating matrices on grids with arbitrarily spaced points. It is shown that high accuracy can be achieved through use of differentiating matrices on non-uniform grids through the expedient of including 'near boundary' points. Use of the differentiating matrix as an operator in the solution of problems involving ordinary differential equations is also considered.
A robust self-starting explicit architecture for computational structural dynamics is described. The proposed methodology involves expressing the governing equations of motion in conservation form and temporal discretization is accomplished in the spirit of the Lax-Wendroff type formulations. The development of the basic methodology is shown. Discretization in space is accomplished by introducing stress-based representations and employing the classical Galerkin scheme. Numerical test model results are presented which validate the architecture.
We consider the efficient numerical solution of the three-dimensional wave equation with Neumann boundary conditions via time-domain boundary integral equations. A space-time Galerkin method with $C^\infty$-smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time.
Excitations of disordered systems such as glasses are of fundamental and practical interest but computationally very expensive to solve. Here we introduce a technique for modeling these excitations in an infinite disordered medium with a reasonable computational cost. The technique relies on a discrete atomic model to simulate the low-energy behavior of an atomic lattice with molecular impurities. The interaction between different atoms is approximated using a spring like interaction based on the Lennard Jones potential but can be easily adapted to other potentials. The technique allows to solve a statistically representative number of samples with a minimum of computational expense, and uses a Monte-Carlo approach to achieve a state corresponding to any given temperature. This technique has already been applied successfully to a problem with interest in condensed matter physics: the solid solution of N$_2$ in Ar.
Plate finite elements based on the generalized third-order theory of Reddy and the first-order shear deformation theory are analyzed and compared on the basis of thick and thin plate modeling behavior, distortion sensitivity, overall accuracy, reliability, and efficiency. In particular, several four-noded Reddy-type elements and the nine-noded Lagrangian and heterosis (Mindlin-type) plate elements are analyzed to assess their behavior in bending, vibration, and stability of isotropic and laminated composite plates. A four-noded Reddy-type element is identified which is free of all spurious stiffness and zero energy modes, computationally efficient, and suitable for use in any general-purpose finite element program.
An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or 'drilling' degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not 'lock', possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.
An improved four-node quadrilateral assumed-stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger-Reissner variational principle and the shape functions are formulated directly for the four-node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has nine independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling, and linear free vibration problems. The element is validated with standard tests cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.
A new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.
A simple computational procedure is presented for reducing the size of the analysis model for a symmetric structure with asymmetric boundary conditions to that of the corresponding structure with symmetric boundary conditions. The procedure is based on approximating the asymmetric response of the structure by a linear combination of symmetric and antisymmetric global approximation vectors (or modes). The key elements of the procedure are (a) restructuring the governing finite element equations to delineate the contributions to the symmetric and antisymmetric components of the asymmetric response, (b) successive application of the finite element method and the classical Rayleigh–Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh–Ritz technique. A tracing parameter is introduced which identifies all the contributions to the antisymmetric response. The global approximation vectors are selected to be the solution corresponding to a zero value of the tracing parameter and the various-order derivatives of the solution with respect to this parameter, evaluated at zero value of the parameter. The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions. The effectiveness of the computational procedure is demonstrated by means of numerical examples of linear static problems of shells, and its potential for solving non-linear problems is discussed.
In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi-analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended scaled boundary finite element method (xSBFEM). Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple MATLAB code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code.
A new scheme to integrate a system of stiff differential equations for both the elasto-plastic creep and the unified viscoplastic theories is presented. The method has high stability, allows large time increments, and is implicit and iterative. It is suitable for use with continuum damage theories. The scheme was incorporated into MARC, a commercial finite element code through a user subroutine called HYPELA. Results from numerical problems under complex loading histories are presented for both small and large scale analysis. To demonstrate the scheme's accuracy and efficiency, comparisons to a self-adaptive forward Euler method are made.
The use of multiple-time-step integrators can provide substantial computational savings over traditional one-time-step methods for the simulation of solid dynamics, while maintaining desirable properties, such as energy conservation. Contact phenomena generally require the adoption of either an implicit algorithm or the use of unacceptably small time steps to prevent large amount of numerical dissipation from being introduced. This paper introduces a new explicit dynamic contact algorithm that, by taking advantage of the asynchronous time-stepping of Asynchronous Variational Integrators (AVI), delivers an outstanding energy performance at a much more acceptable computational cost. We demonstrate the performance of the numerical method with several three-dimensional examples.
A simple but nontrivial steady-state creeping elasto visco-plastic (Maxwell fluid) radial flow problem is analyzed, with special attention given to the effects of the boundary conditions. Solutions are obtained through integration of a governing equation on stress using the Runge-Kutta method for initial value problems and finite differences for boundary value problems. A more general approach through the finite element method, an approach that solves for the velocity field rather than the stress field and that is applicable to a wide range of problems, is presented and tested using the radial flow example. It is found that steady-state flows of elasto visco-plastic materials are strongly influenced by the state of stress of material as it enters the region of interest. The importance of this boundary or initial condition in analyses involving materials coming into control volumes from unusual stress environments is emphasized.
Top-cited authors
Nicolas Moes
  • Ecole Centrale de Nantes
R. L. Taylor
  • University of California, Berkeley
Charbel Farhat
  • Stanford University
Jean-François Remacle
  • Université Catholique de Louvain - UCLouvain
Christophe Geuzaine
  • University of Liège